This is the first lesson in the Unit on Sound.
It introduces the concept of a Restoring Force, and systems with motion subject to such a force.
This motion is repeating, retracing the same path over and over, which is different from either the linear motion, or the two dimensional projectile motion, that students have already studied.
This motion, known as Harmonic Motion, has many real world examples, which the students will recognize.
In many real world examples, both jobs are combinedd as one part; a guitar string is both the Mass and the Spring. We call that a distributed system. It is easier to learn with a separate Mass and Spring. We call that a discrete system
This lesson uses two simple discrete mechanical systems that have Harmonic Motion.
Students will observe their similarities and differences, and may be surprised by some of what they learn. Don't tell them what the surprises are.
Opportunities to Learn:
Students readily understand, and can confirm by experiment, that a heavier object will tend to move more slowly, and thus retraces a path at a lower frequency, while increasing the restoring force (such as the tension on a spring) will increase the frequency. But there is a surprise coming. This is the first learning opportunity.
The fact that the energy we provided to start the system moving continually cycles between elastic and kinetic, is not obvious, but easily understood once it is pointed out.
In any real system, this energy is gradually lost, to friction, or by some of the motion being transferred to the surroundings.
The range of motion, or Amplitude, decreases as energy is lost. But there is another surprise coming. As the Amplitude gradually decreases, the original frequency is maintained, again not obvious, but easily seen once it is pointed out. How can it do that?
Differentiation
The advanced portion of this lesson uses, and may serve as an introduction to, the idea of a State Diagram,
and analysis based on the System State
These relations are derived from first principles, using Newton's laws of motion.
They are then shown both in algebra and graphic form.
This analysis is another learning opportunity, both to learn more about Harmonic Motion, and more importantly, to gain insight into the power of analysis to reveal a deeper understanding of a system.
Depth of Knowledge
Dok1 and Dok2
This lesson introduces new terms and concepts, and gives students hands-on examples to work with, and observe while varying certain parameters.
Students are asked to note which parameters caused changes in system behavior, and to note the similarities and differences of the different example systems.
This is followed by a lecture in which the mechanical systems and the observations are compared.
Prerequisite Knowledge
Students should know and understand Newton's laws of motion.
Every body remains in a state of rest or uniform motion unless it is acted upon by an external unbalanced force.
A body subject to a force undergoes an acceleration proportional to the force, divided by the mass of the body.
The forces betwen two bodies are equal and opposite.
Plans for Differentiating Instruction
Pre-Lab handout for every lab exercise. Needed or not, these include Graphic Organizers, a Glossary of new terms.
Pre-Lab also has a Readiness Test that must be correctly completed as a ticket to do the lab. (but see below)
Analysis in the following lecture is presented in graphical as well as algebraic form. (advanced)
Accommodations and modifications
This lesson begins with laboratory exercises, which are intended to be done by small teams. The instructor may want to form teams in which a student with a special need may be teamed with students that can provide needed assistance.
Lab worksheets for data collection ordinarily start empty, with only required sections indicated, but customized sheets can be provided to give additional guidance.
Pre-Lab Readiness Test may be waived if the student will have assistance and direct supervision.
Environmental factors
None
Materials
The Spring and Mass setup requires a lab stand capable of holding a support point about a meter above the base, a spring, , a way of hanging weights (masses) at the bottom, and several known masses.
The Pendulum setup requires a lab stand capable of holding a support point about a meter above the base, a string or wire, an attachment at the top that allows the length of the string or wire to be adjusted, a way of hanging weights (masses) at the bottom, and several known masses.
If possible, one setup for each lab should be instrumented with e.g. Vernier Labview or Pasco Probeware so nthat students can see the graphs develop as the experimental system operates. This could best be used as part of the lecture. It would also be good as the attention getter for the opening remarks. It is not needed for every student station, but could be useful for additional support as needed.
Objectives:
Students will have worked with several examples of SHM systems.
Students will have observed, and can explain, which parameters may be changed to modify the functioning of an SHM system.
Students can explain the similarities and differences between these SHM systems.
Instruction:
Opening:
Have a lab setup where everyone can see it, and use it for your introduction.
Put both the Spring Mass and the Pendulum in motion.
If you have an instrumented setup, this is the place for it.
Let them continue while you take attendance and then make opening remarks.
This lesson studies Harmonic Motion with two discrete systems:
When you wortk with them you will find some things similar about both, some things different about both, and something that will probably surprise you.
You will have to observe and record both frequency and displacement over time.
You will be able to experiment with some factors that may influence the frequency.
Pendulum Motion
You canl vary the mass, the length of the pendulum, and the starting displacement.
Mass on a Spring
You canl vary the mass, and the spring stiffness, and the starting displacement. (Vary the spring stiffness by using a different spring. You can determine the actual stiffness by first noting the length with no mass attached, then adding the mass and noting the increase in length.)
Procedure
When you move to a lab, start by filling in on your work sheet, a diagram with labels showing important parts.
Then determine what can be varied, and what values you want to try, and draw a suitable table for recording data.
Finally, take some data. Remember to vary one thing at a time, to avoid confusion.
Words of Caution
Pendulum should only move through small angles, never more than twenty degrees from the vertical.
Mass on a Spring should not be extended more than the working range of the spring. The way to be sure is this:
Notice how much the spring stretched by hanging the mass on it.
Never pull it down by more than that same distance further.
This guarantees that when the spring moves to the maxiumum upward position, it will still be visibly stretched, and providing the support for the mass.
(Typically that means the initial downward stretch is no more than fifteen centimeters.)
Send students to begin their labs. Assign half to start with a Pendulum and half to start with a Mass on a Spring.
Engagement:
Students set up their work sheets and begin their lab work.
Walk around to help as needed, and to make sure nothing is done that is not directly called for, or could cause damage.
As time permits, you may want to have them stop and trade positions at a half-way time, but it is not strictly necessary, because there will be enough data to share.
Closure:
We learned that seemingly different mechanical systems have similar behavior, but some differences.
We observed that any such system has a stable position, where it has minimum energy.
If energy is applied to move the system away from the stable position, the restoring force will act to return it to that position.
But when the stable position is reached, the system will be in motion, and momentum will carry it beyond the stable position.
From there, the restoring will again act, and the cycle will repeat
What was the same? Repeating motion at a steady frequency, with gradual return to the stable position.
What was different?
Mass on Spring frequency depended on both mass and spring stiffness.
Pendulum frequency did not depend on mass, only length of support. Why?
What was surprising? Two surprises:
Frequency remained constant as amplitude decreased.
Pendulum frequency did not depend on mass.
Advanced
. (This section is still rough, needs clearer explanation and better notation, and it needs some graphs; I'm still learning to draw them. It is here as a place holder for the advanced section of Differentiated Instruction.)
In the analysis of a system, such as the vibrating string, or mass on a spring, or a pendulum, we can introduces the concept of a System State, described in terms of the object displacement vs object speed, since those are two values we can observe and measure.
Now we can draw a State Diagram, a graph in which the axes are object displacement vs object speed.
Note!
Be sure to label the axes and point out to the class that they do not use the same units.
Any point drawn on this graph will represent a particular combination of object displacement vs object speed.
This is different from other graphs the students have seen, such as X vs Time in one-dimensional motion, or X vs Y for two-dimensional projectile motion.
Those were all actual coordinates of the time and physical space holding the system, but here the axes are parameters of the system, that we have chosen to help our analysis.
With a suitable scaling constant, we can make the maximum extent on the two axes have the same length.
We know that the system has two forms of energy; potential energy due to the displacement against the restoring force, and kinetic energy due to the speed.
Furthermore, we can apply Newton's laws to show that any point on the graph represents:
potential energy proportional to the square of the displacement against the restoring force,
kinetic energy proportional to the square of the speed.
These energy equations should be derived as a review of prior knowledge, and to bring them into current working knowledge. They can be derived using both algebraic and graphical representation, with explanation linking them together. No calculus is needed, although it would be if the forces were more complicated.
Potential energy developed against a constant force, such as gravity, is easily seen to be Force * Distance, or represented in a graph as a rectangular area.
For a linear restoring force, we notice that the force is the distance x k, the spring constant, thus starts at zero, has a maxiumum of Distance x k, and an average of half that. Multiplying by Distance gives k/2 x Distance^^2. The graphic representation is a triangular area.
Kinetic energy developed by a constant force, such as gravity, is again seen to be Force * Distance, during which time the constant acceleration causes the speed to increase linearly, with an average value of half the maximum. Thus it is m/2 x Velocity^^2.
Kinetic energy developed by a linear force, and thus a linearly increasing acceleration, has an average acceleration of half the maximum, which leads to the same result of m/2 * Velocity^^2 although actual plot of velocity vs time would be different. (This is in itself worth mentioning.)
We understand that the sum of these two forms of energy must be a constant, as long as we assume no friction or other losses.
So there are places on the graph where this sum is a constant, and they are "allowed" states of the system.
And other places are not allowed states.
We can then apply the Pythagoean Theorem to show that a circle represents all possible combinations of speed and displacement that could correspond to a constant total energy, conveniently shown as the square of the radius of the circle.
Thus, the circle shows all possible energy values for the system, while a point traveling around the circle describes one cycle of the vibration. If we draw a line (a radius) from the center to the point, the angle it makes is a convenient description of where we are in the cycle. From the definition of sine and cosine, we can see that the sine of this angle gives the current velocity, while the cosine gives the current displacement.
Again, be sure to point out that "nothing is moving in a circle" except the point in the State Diagram.
This analysis began by assuming no energy losses, and a motion that continues forever. To account for energy loss, due to friction, or to generate a sound, for example, the point no longer moves in a circle; it spirals inward to the center, which represents a stable state with no displacement and no speed.
Assesment:
Students answer these questions after doing the pre-lab reading, and before the lab.
Choose the best definition for each.
(1) Newton's First Law
(2) Newton's Second Law
(3) Newton's Second Law
(4) Stable Position of a mechanical system
(5) Hooke's Law for Springs
(6) Restoring Force in a mechanical system
(7) Potential Energy of a mechanical system
(8) Kinetic Energy of a mechanical system
(9) Natural Frequency of a mechanical system
(10) Resonance of a mechanical system
(A) a body remains at rest or in motion with a constant velocity unless acted upon by an external force.
(B) a heavier object will fall more quickly than a lighter object.
(C) the rate of change of momentum is proportional to the imposed force and goes in the direction of the force.
(D) a periodic motion in which the displacement is symmetrical about a point
(E) the configuration in which a system comes to rest
(F) mechanical energy that a body has by virtue of its motion
(G) the specific frequency at which a system started into motion vibrates
(H) stored energy that a body has by virtue of its position
(I) the principle that the change in size of a solid is proportional to the force applied to it
(J) a force always directed towards the stable position.
(K) action and reaction are equal and opposite.
Section II: Understanding
As shown by individual (not team) lab reports, and participation in discussion after the lab.
Lab reports are looked at by the teacher, but stay with students for use as study guides.
Rhode Island Department of Education Lesson Plan
Lesson Title: Motion With a Restoring Force
State Standards: GLEs/GSEs
Listed in Unit plan
National Standards:
Listed in Unit plan
Context of Lesson:
This is the first lesson in the Unit on Sound.
It introduces the concept of a Restoring Force, and systems with motion subject to such a force.
This motion is repeating, retracing the same path over and over, which is different from either the linear motion, or the two dimensional projectile motion, that students have already studied.
This motion, known as Harmonic Motion, has many real world examples, which the students will recognize.
In many real world examples, both jobs are combinedd as one part; a guitar string is both the Mass and the Spring. We call that a distributed system. It is easier to learn with a separate Mass and Spring. We call that a discrete system
This lesson uses two simple discrete mechanical systems that have Harmonic Motion.
Students will observe their similarities and differences, and may be surprised by some of what they learn. Don't tell them what the surprises are.
Opportunities to Learn:
Differentiation
The advanced portion of this lesson uses, and may serve as an introduction to, the idea of a State Diagram,
and analysis based on the System State
These relations are derived from first principles, using Newton's laws of motion.
They are then shown both in algebra and graphic form.
This analysis is another learning opportunity, both to learn more about Harmonic Motion, and more importantly, to gain insight into the power of analysis to reveal a deeper understanding of a system.
Depth of Knowledge
Dok1 and Dok2
This lesson introduces new terms and concepts, and gives students hands-on examples to work with, and observe while varying certain parameters.
Students are asked to note which parameters caused changes in system behavior, and to note the similarities and differences of the different example systems.
This is followed by a lecture in which the mechanical systems and the observations are compared.
Prerequisite Knowledge
Students should know and understand Newton's laws of motion.
Plans for Differentiating Instruction
Accommodations and modifications
Environmental factors
None
Materials
The Spring and Mass setup requires a lab stand capable of holding a support point about a meter above the base, a spring, , a way of hanging weights (masses) at the bottom, and several known masses.
The Pendulum setup requires a lab stand capable of holding a support point about a meter above the base, a string or wire, an attachment at the top that allows the length of the string or wire to be adjusted, a way of hanging weights (masses) at the bottom, and several known masses.
If possible, one setup for each lab should be instrumented with e.g. Vernier Labview or Pasco Probeware so nthat students can see the graphs develop as the experimental system operates. This could best be used as part of the lecture. It would also be good as the attention getter for the opening remarks. It is not needed for every student station, but could be useful for additional support as needed.
Objectives:
Instruction:
Opening:
Have a lab setup where everyone can see it, and use it for your introduction.
Put both the Spring Mass and the Pendulum in motion.
If you have an instrumented setup, this is the place for it.
Let them continue while you take attendance and then make opening remarks.
This lesson studies Harmonic Motion with two discrete systems:
When you wortk with them you will find some things similar about both, some things different about both, and something that will probably surprise you.
You will have to observe and record both frequency and displacement over time.
You will be able to experiment with some factors that may influence the frequency.
- Pendulum Motion
You canl vary the mass, the length of the pendulum, and the starting displacement.- Mass on a Spring
You canl vary the mass, and the spring stiffness, and the starting displacement. (Vary the spring stiffness by using a different spring. You can determine the actual stiffness by first noting the length with no mass attached, then adding the mass and noting the increase in length.)Procedure
Words of Caution
Send students to begin their labs. Assign half to start with a Pendulum and half to start with a Mass on a Spring.
Engagement:
Students set up their work sheets and begin their lab work.
Walk around to help as needed, and to make sure nothing is done that is not directly called for, or could cause damage.
As time permits, you may want to have them stop and trade positions at a half-way time, but it is not strictly necessary, because there will be enough data to share.
Closure:
We learned that seemingly different mechanical systems have similar behavior, but some differences.
We observed that any such system has a stable position, where it has minimum energy.
If energy is applied to move the system away from the stable position, the restoring force will act to return it to that position.
But when the stable position is reached, the system will be in motion, and momentum will carry it beyond the stable position.
From there, the restoring will again act, and the cycle will repeat
Advanced
.
(This section is still rough, needs clearer explanation and better notation, and it needs some graphs; I'm still learning to draw them. It is here as a place holder for the advanced section of Differentiated Instruction.)
In the analysis of a system, such as the vibrating string, or mass on a spring, or a pendulum, we can introduces the concept of a System State, described in terms of the object displacement vs object speed, since those are two values we can observe and measure.
Now we can draw a State Diagram, a graph in which the axes are object displacement vs object speed.
Note!
Be sure to label the axes and point out to the class that they do not use the same units.
Any point drawn on this graph will represent a particular combination of object displacement vs object speed.
This is different from other graphs the students have seen, such as X vs Time in one-dimensional motion, or X vs Y for two-dimensional projectile motion.
Those were all actual coordinates of the time and physical space holding the system, but here the axes are parameters of the system, that we have chosen to help our analysis.
With a suitable scaling constant, we can make the maximum extent on the two axes have the same length.
We know that the system has two forms of energy; potential energy due to the displacement against the restoring force, and kinetic energy due to the speed.
Furthermore, we can apply Newton's laws to show that any point on the graph represents:
These energy equations should be derived as a review of prior knowledge, and to bring them into current working knowledge. They can be derived using both algebraic and graphical representation, with explanation linking them together. No calculus is needed, although it would be if the forces were more complicated.
We understand that the sum of these two forms of energy must be a constant, as long as we assume no friction or other losses.
So there are places on the graph where this sum is a constant, and they are "allowed" states of the system.
And other places are not allowed states.
We can then apply the Pythagoean Theorem to show that a circle represents all possible combinations of speed and displacement that could correspond to a constant total energy, conveniently shown as the square of the radius of the circle.
Thus, the circle shows all possible energy values for the system, while a point traveling around the circle describes one cycle of the vibration. If we draw a line (a radius) from the center to the point, the angle it makes is a convenient description of where we are in the cycle. From the definition of sine and cosine, we can see that the sine of this angle gives the current velocity, while the cosine gives the current displacement.
Again, be sure to point out that "nothing is moving in a circle" except the point in the State Diagram.
This analysis began by assuming no energy losses, and a motion that continues forever. To account for energy loss, due to friction, or to generate a sound, for example, the point no longer moves in a circle; it spirals inward to the center, which represents a stable state with no displacement and no speed.
Assesment:
Students answer these questions after doing the pre-lab reading, and before the lab.
Choose the best definition for each.
(1) Newton's First Law
(2) Newton's Second Law
(3) Newton's Second Law
(4) Stable Position of a mechanical system
(5) Hooke's Law for Springs
(6) Restoring Force in a mechanical system
(7) Potential Energy of a mechanical system
(8) Kinetic Energy of a mechanical system
(9) Natural Frequency of a mechanical system
(10) Resonance of a mechanical system
(A) a body remains at rest or in motion with a constant velocity unless acted upon by an external force.
(B) a heavier object will fall more quickly than a lighter object.
(C) the rate of change of momentum is proportional to the imposed force and goes in the direction of the force.
(D) a periodic motion in which the displacement is symmetrical about a point
(E) the configuration in which a system comes to rest
(F) mechanical energy that a body has by virtue of its motion
(G) the specific frequency at which a system started into motion vibrates
(H) stored energy that a body has by virtue of its position
(I) the principle that the change in size of a solid is proportional to the force applied to it
(J) a force always directed towards the stable position.
(K) action and reaction are equal and opposite.
Section II: Understanding
As shown by individual (not team) lab reports, and participation in discussion after the lab.
Lab reports are looked at by the teacher, but stay with students for use as study guides.